$ F = \left[\begin{array}{rrr}1 & -2 & -1 \\ -1 & 0 & 0\end{array}\right]$ $ D = \left[\begin{array}{rr}-2 & 3 \\ 3 & 5 \\ 2 & -1\end{array}\right]$ What is $ F D$ ?
Solution: Because $ F$ has dimensions $(2\times3)$ and $ D$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ F D = \left[\begin{array}{rrr}{1} & {-2} & {-1} \\ {-1} & {0} & {0}\end{array}\right] \left[\begin{array}{rr}{-2} & \color{#DF0030}{3} \\ {3} & \color{#DF0030}{5} \\ {2} & \color{#DF0030}{-1}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ F$ , with the corresponding elements in column $j$ of the second matrix, $ D$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ F$ with the first element in ${\text{column }1}$ of $ D$ , then multiply the second element in ${\text{row }1}$ of $ F$ with the second element in ${\text{column }1}$ of $ D$ , and so on. Add the products together. $ \left[\begin{array}{rr}{1}\cdot{-2}+{-2}\cdot{3}+{-1}\cdot{2} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ F$ with the corresponding elements in ${\text{column }1}$ of $ D$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{-2}+{-2}\cdot{3}+{-1}\cdot{2} & ? \\ {-1}\cdot{-2}+{0}\cdot{3}+{0}\cdot{2} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ F$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ D$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{-2}+{-2}\cdot{3}+{-1}\cdot{2} & {1}\cdot\color{#DF0030}{3}+{-2}\cdot\color{#DF0030}{5}+{-1}\cdot\color{#DF0030}{-1} \\ {-1}\cdot{-2}+{0}\cdot{3}+{0}\cdot{2} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{1}\cdot{-2}+{-2}\cdot{3}+{-1}\cdot{2} & {1}\cdot\color{#DF0030}{3}+{-2}\cdot\color{#DF0030}{5}+{-1}\cdot\color{#DF0030}{-1} \\ {-1}\cdot{-2}+{0}\cdot{3}+{0}\cdot{2} & {-1}\cdot\color{#DF0030}{3}+{0}\cdot\color{#DF0030}{5}+{0}\cdot\color{#DF0030}{-1}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-10 & -6 \\ 2 & -3\end{array}\right] $